The impasse has resulted in the furlough of some 4,000 federal workers

Washington (CNN) -- Congressional leaders reached a deal Thursday to temporarily resume funding for the Federal Aviation Administration, ending an impasse that put thousands of federal employees out of work and halted activity on 200 airport construction projects.

President Barack Obama hailed the agreement that he said allows tens of thousands of people to go back to work.

"We can't afford to let politics in Washington hamper our recovery, so this is an important step forward," Obama said in a prepared statement.

For workers facing the possibility of lost wages for weeks to come, news of the breakthrough evoked an emotional response. Evelyn Martinez, an FAA inspector, started crying when she learned of the deal while being interviewed by CNN.

"They're out of relief," Martinez said of her tears, adding that while her post-graduate education taught her how government works, "what we are experiencing is outside of any textbook."

According to a Senate Democratic leadership aide, the Senate will pass the necessary legislation by unanimous consent on Friday in a so-called "pro forma" session being held while the chamber is in recess. Majority Leader Harry Reid, as the lone senator present, will conduct the pro forma session and call for unanimous consent to pass the measure, which previously was approved by the House, the aide said on condition of not being identified.

The impasse has resulted in the furlough of roughly 4,000 aviation workers, as well as tens of thousands of additional layoffs in the construction industry and elsewhere. With Congress on recess until September, the possibility of workers being idled for weeks at the height of the construction season further angered an American public already frustrated by what it considers unnecessary political squabbling in Washington.

Since authorization for FAA funding expired in late July, the agency has been unable to collect federal taxes on airline tickets -- leading to a revenue loss of approximately $30 million a day. If the dispute continued until Congress returned from its summer recess in September, the federal government would have been out more than $1 billion in revenue.

Reid has been in talks since Thursday with House Speaker John Boehner and the Obama administration about how to resolve the FAA funding impasse, the Senate Democratic leadership aide said.

Earlier, White House Press Secretary Jay Carney told reporters that Obama called Boehner, R-Ohio, on Wednesday to discuss ways to resolve the matter.

"This is a tremendous victory for American workers everywhere," Transportation Secretary Ray LaHood said in a statement on the deal. "From construction workers to our FAA employees, they will have the security of knowing they are going to go back to work and get a paycheck -- and that's what we've been fighting for."

In his own statement, Reid, D-Nevada, noted that the agreement "does not resolve the important differences that still remain."

"But I believe we should keep Americans working while Congress settles its differences, and this agreement will do exactly that," Reid's statement said.

LaHood had urged members of Congress to return to Washington from their summer break and, at a minimum, pass a temporary funding measure allowing the FAA and other workers to return to their jobs.

The Democratic-led Senate went on its summer recess Tuesday without approving what would have been the 21st short-term funding extension for the FAA. The Republican-led House previously passed a short-term extension, but included some changes opposed by Democrats.

The dispute over the extension involves language in the House proposal that would reduce or kill subsidies to rural airports, specifically targeting some in Nevada, Montana and New Mexico -- three states with influential Democratic senators.

As part of Thursday's agreement, LaHood will notify congressional leaders that he has the authority to grant waivers to affected community airports, which would effectively exempt them from the subsidy cuts, according to the Senate Democratic leadership aide. While no specific exemptions are required as part of the deal, the authority held by LaHood gives Democrats the assurance that the disputed language they opposed can be overridden, the aide noted.

A larger dispute behind the scenes also was a cause for the inaction, and the deal announced Thursday does not address it.

Republicans oppose a recent National Mediation Board decision backed by Democrats that makes it easier for airline employees to unionize.

The board's ruling made passage of a vote to unionize dependent on getting more than 50% support of those voting. For example, if a company has 1,000 workers but only 200 take part in the vote to unionize, the rule change would meant only 101 "yes" votes were needed for it to pass.

Under old rules, more than 50% of all workers eligible to vote -- or in this example 501 "yes" votes -- would have been required for it to pass. Workers who didn't cast ballots were counted as having voted "no," making it more difficult for supporters to succeed.

In statements Thursday responding to the deal, some Democratic legislators continued to blame Republicans for the impasse.

"It's clear the right wing of the GOP wants to undo worker protections and may again block progress on the FAA bill in September in order to get its way," Sen. Jay Rockefeller, D-West Virginia, said in a statement. "Thankfully, for now, this deal allows the FAA to restart, maintains workers' rights, and ensures that rural airports can get the resources they need, with language that protects deserving small communities whose airports are the lifeblood of their economies."

LaHood said Thursday morning, before the deal was announced, that controversial items didn't stop lawmakers from passing extensions on other occasions.

"If you've got issues with labor, if you've got issues with money going to small airports to help airlines fly in and out, work that out," he said. "Don't hold the American jobs and American people hostage over controversial issues that were not a problem on 20 other times when Congress passed an extension."

As the dispute dragged on, numerous FAA employees were being forced to dig into personal savings, prioritize their bills, and cut back on expenses in order to avoid financial devastation.

"We're pretty much going to burn through all of our savings within a month and ... now we're working on programs out there to give us no-interest loans," said Mark DePlasco, one of the furloughed employees. "I don't think any of us can even fathom going without a paycheck for another month and a half or even longer."